The 3-SvPS reconfigurable parallel mechanism (Ye, Chai and Zhang, 2020) attains four mobility configurations with 6, 5, 4 and 3 degrees of freedom by switching each of its three Sv-P-S limbs' metamorphic spherical variable-axis joint between a universal-like and a revolute-like phase. Its singularities were characterized only numerically: the authors derived a unified screw-theoretic Jacobian valid in every configuration and mapped the forward-kinematic singular sets det⁡(Jx)=0\det(J_x)=0 by evaluating the determinant on two-dimensional pose slices for one nominal architecture. This record formulates the analytic problem that work left open: derive closed-form singular-surface equations for each of the four configurations as explicit functions of the pose variables and the design radii aa and bb, with a decomposition into transmission and constraint components. No closed-form singular-surface analysis for this mechanism or for the wider family of variable-axis-joint reconfigurable parallel mechanisms is known; the problem remains open.

Background

Parallel mechanisms are robotic manipulators in which a moving platform is connected to a fixed base by several kinematic chains (limbs) acting in parallel; this multi-loop topology gives high stiffness and lets the actuators sit on the base, at the cost of a smaller workspace and of singular poses at which the mechanism loses controllability. A reconfigurable parallel mechanism (RPM) can change its effective topology — and hence its number of degrees of freedom (DOF) — so that one machine can serve tasks with different motion requirements. One route to reconfigurability is the metamorphic joint, a joint whose kinematic behaviour changes when part of it is locked or reoriented. Ye, Chai and Zhang (2020) study an RPM built from such joints: three identical limbs of type Sv-P-S, where P is an actuated prismatic (sliding) joint, S is a spherical (ball-and-socket) joint, and Sv is a metamorphic spherical variable-axis joint (the vA joint) that switches between a Uv phase, behaving like a universal joint and adding an actuated revolute (hinge) axis R1R_1, and an Rv phase, behaving like a single revolute joint whose other axis is locked. Because each of the three Sv joints can be switched independently, the 3-SvPS mechanism realizes four mobility configurations with 6, 5, 4 and 3 DOF — 3-UvPS, 2-UvPS-1-RvPS, 1-UvPS-2-RvPS and 3-RvPS (with 3R2T, 3R1T and 2R1T denoting three-rotation-two-translation, three-rotation-one-translation and two-rotation-one-translation motion) — each exerting a different set of constraint wrenches (force–moment systems) on the platform.

Singularity analysis of parallel mechanisms is conventionally posed through the Jacobian relation Jx δp=Jq q˙J_x\,\delta_p=J_q\,\dot{q} between the platform twist (instantaneous velocity state) and the actuated joint rates, constructed with screw theory, the representation of instantaneous twists and wrenches as screws in which reciprocity expresses ideal power transmission. Forward-kinematic (parallel) singularities occur where det⁡(Jx)=0\det(J_x)=0, i.e. where the transmission and constraint wrench screws acting on the platform become linearly dependent; at such poses the platform gains uncontrollable motion even with all actuators locked. These are distinct from serial (actuation) singularities associated with JqJ_q, and are related to the constraint singularities classified by Zlatanov, Bonev and Gosselin (2002) — poses where the platform's constraint wrench system degenerates and the mobility itself changes. For fixed-architecture mechanisms, closed-form singularity analyses are well developed, for example the geometric-algebra derivation of the 3-RPS manipulator's singularity variety (the set of singular poses expressed as an algebraic surface in pose coordinates) by Li, Xiang, Chai and Wu (2015), and the analytical singularity-free tubes for the 6-6 Stewart platform derived by Kolte, Ramesh and Bandyopadhyay (2024). For mechanisms with multiple operation modes, per-mode treatments exist for specific architectures: Gan, Dai, Dias and Seneviratne (2013) gave a unified kinematics and singularity analysis of a metamorphic mechanism with bifurcated motion, and the same authors later derived analytic singularity loci and a singularity-free workspace for a metamorphic parallel mechanism built from reconfigurable Hooke (rT) joints with controllable rotation center (2014); Li, Feng and Dai (2026) gave a Grassmann-Cayley-algebra-based singularity analysis of a 2-URU (two universal-revolute-universal-limb) metamorphic mechanism; Chablat, Kong and Zhang (2018) analyzed a robot with five operation modes; Nurahmi and Gan (2019) studied reconfiguration of the 3-(rR)PS metamorphic mechanism from workspace and operation-mode analysis, and Nurahmi, Gan and Setya (2023) later crossed its singularities through dynamic modeling rather than analytically characterizing them. These analytic or semi-analytic treatments all concern reconfigurable-joint architectures other than the variable-axis (Sv) joint, and none derives closed-form singular-surface equations carrying the design parameters symbolically for a multi-configuration mechanism.

For the 3-SvPS mechanism itself, Ye, Chai and Zhang (2020) derived a unified Jacobian JxJ_x, assembled from the reciprocal transmission and constraint wrench screws, valid in all four mobility configurations, and used it together with the motion/force transmissibility indices of Wang, Wu and Liu (2010) (dimensionless local and global measures of how well actuator effort converts to platform motion) for a two-parameter design optimization over the base radius aa and platform radius bb. Their singularity results, however, are numerical and slice-based: for the nominal architecture a=100a=100 mm and b=50b=50 mm, the singular sets det⁡(Jx)=0\det(J_x)=0 were located by evaluating the determinant on two-dimensional pose slices (a zero-orientation translation slice and a fixed-height orientation slice) and interpolating the resulting curves, which revealed that interior singular surfaces partition the workspace in some modes. No closed-form equations of the singular surfaces carrying the design radii symbolically were given — this is the open problem.

Problem Statement

For the 3-SvPS reconfigurable parallel mechanism of Ye, Chai and Zhang (2020), establish a closed-form analytic singularity characterization covering all four mobility configurations: for each configuration — 3-UvPS with 6 DOF, 2-UvPS-1-RvPS with 5 DOF, 1-UvPS-2-RvPS with 4 DOF, and 3-RvPS with 3 DOF — derive explicit closed-form singular-surface equations in the platform pose variables, position (x,y,z)(x,y,z) and Z-Y-X Euler angles (α,β,γ)(\alpha,\beta,\gamma), with the base radius aa and platform radius bb carried symbolically; that is, express det⁡(Jx)\det(J_x) of the unified screw-theoretic Jacobian published in the source paper as a product of explicit closed-form factors, not merely the unreduced determinant, such that each factor vanishes exactly at a specific, named dependence among the transmission (actuation) or constraint wrench screws, the product reproduces det⁡(Jx)\det(J_x) up to a stated nonzero factor, and the common solution set of the factors is the forward-kinematic singular set det⁡(Jx)=0\det(J_x)=0. The Jacobian, limb geometry and pose conventions are those defined in Ye, Chai and Zhang (2020), and the four per-configuration analyses must be mutually consistent with that unified formulation. Parametric studies of how the singular varieties move with aa and bb, and criteria certifying prescribed pose regions free of interior singularities across configurations, are natural downstream applications of such equations.

References

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